How to rearrange the angle of existing polygons
There's a little trick, which also helps to make diagonal grids for modeling knurls, nettings, trellises, gratings, etc.




WS Subdivide your mesh 
CtrlE edge menu, Un-subivide  it.


Done.

It's odd that the operations aren't inverses of one another ... Un-subdivide it again, and you're back to where you started. As Mr. Zak point out in his comment, the iterations should be set to 1. Odd numbers of iterations yield diagonal divisons.